Refining the perfect lens

This limit to the resolving power of a lens wasthought to be an immutable law of nature, but

ARTICLE IN PRESSThe wave vectors kx; ky represent a Fourierdecomposition of the object, so that larger values

recent work has shown that this is not the case. Ifwe can amplify the Fourier components whichnormally decay exponentially, they also can maketheir contribution to the image. It was shown thata parallel-sided slab of material with the property

waves which decay exponentially away from theobject and make little or no contribution to theimage. Hence the nest details of the object areeliminated from the image. For this reason thesedecaying components are referred to as the neareld.themselves as imaginary components to e1 and m1: In this talk we shall examine how to minimise the restrictions bystructuring the lens into a series of thin slices. This results in a novel mapping onto an anisotropic system which behaves

like an optical bre bundle, but operating on the near eld.

r 2003 Elsevier B.V. All rights reserved.

Keywords: 78.20.Ci; 42.30.Wb; 78.45.+h

It is common knowledge that a lens will notproduce an image better resolved than thewavelength of light, l; even though the objectradiating light contains details much ner than l:The reason for this is not hard to see. Suppose theaxis of the lens is the z-axis, so that light wavesleaving the object have the form

of these variables correspond to ner details.However, kz only takes real values if

o2

c202pl

> k2x k2y; 3delivered to the image plane with the correct amplitude and phase reproducing every detail in the original source [Phys.

and suspended in vacuo has the ability to focus a perf/j.physb.2003.08.014332

erfect lens

tha Ramakrishna

ce Consort Road, London SW7 2BU, UK

mage: both the near-eld and far-eld components are

In the near-eld limit the amplication by theslab is given by

limk2xk2y-N

TP

4e1exp k2x k2y

qd

e1 12 e1 1

2exp 2k2x k2y

qd

: 6

In the ideal situation when e 1; all waves areamplied exponentially

limk2xk2y-Ne-1

TP exp k2x k2y

qd

; 7

but if there is some absorption present the best wecan do is

e1 1 id; 8

then

TPEexp

k2x k2y

qd

d2 exp 2k2 k2

qd

9

ARTICLE IN PRESS

(b) this antisurface plasmon state cannot exist in isolation

because the wave eld diverges at innity; (c) however, surface

and antisurface plasmon combine to give the resonant

amplifying state when two surfaces and an object are present.

a / Pof negative refractive index implied by

e 1; m 1 4

will do the job of amplifying the near eld [1], aswell as focussing the far eld [2], and hence will intheory form a perfect image. In practice thiscondition can never be completely achievedbecause of imperfections in real materials whichwill always absorb some radiation resulting in asmall imaginary component of both e and m: Anegative refractive index has been demonstratedfor the far eld [3,4]. A popular account can befound in Ref. [5]. Many papers have appeared onthis topic, amongst which [613].In the extreme near-eld limit,

o2

c205k2x k

2y; 5

electric and magnetic elds become independent ofone another so that for an image comprising solelyelectric elds we need only require that e 1 torefocus them. This condition is approximated bysilver at a frequency in the optical region of thespectrum and we can use a slab of silver as anapproximation to the perfect lens for objects muchsmaller than the wavelength.In fact we can understand the amplication

process as the excitation of surface plasmonresonances on the surface of the silver. Incidentelds couple to the surface plasmons to justthe right degree to compensate for the exponentialdecay. Fig. 1 shows the situation. Fig. 1(a) showsa surface plasmon wave eld at a free surface.Note that the electric elds are equal and oppositeon either side of the surface. Fig. 1(b) showsanother situation where the elds match at thesurface, but which does not correspond to a truesurface plasmon because the elds diverge expo-nentially at innity. However, this second set ofwave elds is vital to the functionality of the lensbecause they form a bridge between the decayingelds of the object and the surface resonance onthe far surface. Fig. 1(c) shows a slab of silver witha near-eld source exciting a surface plasmonvia an anti-plasmon. As a result the eldsare amplied to their correct values in the

There is a variant on the slab conguration whichgreatly reduces the impact of absorption. ConsiderFig. 2, on the left-hand side in 2(a) we see theoriginal lens in operation. The lens has a thickness

dielectric function of the slices and e2 that for thespaces between, then

e1z 12e11 e

12 0; ez N: 12

Likewise considering the displacement eld paral-lel to the slices gives

ejj 12 e1 e2 121 1 0: 13

The same result would have been achieved byembedding a set of very ne innitely conductingwires in an insulating matrix with e 0: Put likethis it is not hard to see why the system behaves

ARTICLE IN PRESS

will

uch

J.B. Pendry, S.A. Ramakrishna / Physica B 338 (2003) 329332 331d and the total distance between object and imageis 2d. On the right-hand side in 2(b) we see the lenscut into several pieces and the air gap equallydistributed between these pieces. Provided that thematerial is lossless this makes no difference to theimage, because each slablet makes its owncontribution to amplication resulting in the sameamplitude in the image plane and hence a wellfocussed image.However the distribution of the wave eld is

very different in the two cases. For the distributedlens system nowhere does the amplitude becomeextreme. In contrast the original lens may producesome very large amplitudes indeed. This makes ussuspect that the distributed lens may be lesssusceptible to absorption: see Refs. [1113]. Thisis in fact the case.We can consider the limit of an innitely nely

divided lens cut into many thin slices. In this limitwe can treat the slices as an effective medium.Considering the average electric eld leads us tocalculate the effective dielectric function for adirection normal to the slices, ez: If e1 is the

Fig. 2. An ideal lossless lens can be divided into several slices and

presence of losses the performance of distributed lens degrades meld reduces the effect of loss.like a lens: the potential of an object on onesurface is conducted point by point to the othersurface as though they were connected by wires.Fig. 3 makes the point graphically.As long as the system is loss free, the slices are

equivalent to the slab: both refocus the light over adistance 2d. However, the systems are quitedifferent in the presence of losses. Suppose thate1 1 id then

e1z 12

1

1 id 1

E i

2d; ezE2id

1

ejj 12 1 id 1 i2d: 14

In the near-eld limit dispersion of a wave in ananisotropic system is given by

kz kJ

eJez

rE i

2kJd; 15

therefore in the absence of absorption kz 0; theeld propagates through the slices without changeof amplitude or phase, almost like light passingthrough an optical bre bundle, except that thedetails of the image as now much less wavelength.

still produce a focussed image distance 2d from the object. In the

less rapidly because the generally smaller amplitude of the wave

ARTICLE IN PRESS

ach w

n a m

y poi

J.B. Pendry, S.A. Ramakrishna / Physica B 338 (2003) 329332332Fig. 3. On the left we see a perfect lens cut into many thin slices, e

thin slices like a set of very highly conducting wires embedded i

comes about: the equivalent wires conduct the potential point bIn the presence of absorption this is no longer true,and the image suffers some attenuation by

exp 2ikzdEexp kJdd; 16

giving an effective cutoff in resolution at

Dslice 2p

kJmaxE2pdd: 17

Comparison of Dslice and Dslab shows that Dslice hasa much more favourable dependence on d andtherefore the effects of absorption are much lessfor the equivalent slice system. We illustrate this